This final section is devoted to the completion of the proof of Theorem REF . The proof strategy is as follows. \(X_{x}\)  lies in a large maximal subgroup \(Y\)  of \(X\)  by Theorem REF . Moreover, \(Y\)  is classified in [1]}. We filter the \(Y\) -candidates with respect to \(\left( \Phi _{ef}^{\ast }(p),\left|Y\right|\right) >1\) . Then, by comparing the information contained in [2]} and [3]} on the number of conjugacy classes of the subgroups of \(X\)  isomorphic to \(Y\) , with those contained in Lemma REF , we obtain \(X_{x}=Y\) . Hence, \(X_{x}\)  is a known large maximal subgroup of \(X\) . At this point, we can precisely evaluate the diophantine equation \(k^{2}=[X:X_{x}]\) , and, by combining some of the results contained in [4]} with properties arising from the geometry of the classical groups, we reduce to the case \(X \cong PSL_{3}(3)\) , \(k=12\)  and hence to Cases (3) and (4) of Theorem REF .
